An extension operator on bounded domains and applications

Abstract

In this paper we study a sharp Hardy–Littlewood–Sobolev type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain $\Omega$ and show that if the extension constant for $\Omega$ is strictly larger than the extension constant for the unit ball $B_1$ then extremal functions exist. Using suitable test functions we show that this criterion is satisfied by an annular domain whose hole is sufficiently small. The construction of the test functions is not based on any positive mass type theorems, neither on the nonflatness of the boundary. By using a similar choice of test functions with the Poisson-kernel-based extension operator we prove the existence of an abstract domain having zero scalar curvature and strictly larger isoperimetric constant than that of the Euclidean ball.

Appendix: Regularity

In this section we collect some regularity results, the proofs of which follow from standard arguments.

Lemma 5.1

If $u\in L^{2(n - 1)/(n - 2)}(\partial \Omega )$ and $v\in L^{2^*}(\Omega )$ satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle u(y) = \int _\Omega \frac{a(x)v(x)}{\left| x - y\right| ^{n - 2}}\; \mathrm{d}x &{} y\in \partial \Omega \\ \displaystyle v(x) = \int _{\partial \Omega }\frac{b(y)u(y)}{\left| x- y\right| ^{n - 2}}\; \mathrm{d}S_y &{} x\in \Omega \end{array}\right. } \end{aligned}$$

(5.1)

where $a\in L^\sigma (\Omega )$ for some $\sigma > \frac{n}{2}$ and $b\in L^\tau (\partial \Omega )$ for some $\tau > n -1$ then $u\in L^\infty (\partial \Omega )$ and $v\in L^\infty (\Omega )$.

Lemma 5.2

Let $\Omega \subset \mathbb R^n$ be a smooth bounded domain. The restriction operator $R_2$ given in (2.19) maps $L^\infty (\Omega )$ into $C^{0,1}(\partial \Omega )$ and there is a constant $C = C(n, \Omega )>0$ such that for every $g\in L^\infty (\Omega )$,

$$\begin{aligned} \left| R_2g(y) - R_2g(z)\right| \le C\left\| g\right\| _{L^\infty (\Omega )}\left| y - z\right| \end{aligned}$$

for all $y, z\in \partial \Omega$.

Lemma 5.3

Let $\Omega \subset \mathbb R^n$ be a smooth bounded domain. The restriction operator $R_2$ given in (2.19) maps $L^\infty (\Omega )$ into $C^1(\partial \Omega )$.

Lemma 5.4

If $f\in L^\infty (\partial \Omega )$ then for every $0< \beta <1$, $E_2 f\in C^{0, \beta }({{\overline{\Omega }}})$ and there is a constant $C = C(n, \Omega , \beta )$ such that for all $x, z\in {{\overline{\Omega }}}$

$$\begin{aligned} \left| E_2 f(x) - E_2 f(z)\right| \le C\left\| f\right\| _{L^\infty (\partial \Omega )}\left| x-z\right| ^\beta . \end{aligned}$$